A263008 First member T0(n) of the smallest positive pair (T0(n), U0(n)) for the n-th 2-happy number couple (D(n), E(n)).

1, 1, 1, 3, 1, 13, 1, 1, 5, 7, 1, 1, 3, 59, 1, 1, 7, 23, 1, 221, 7, 1, 1, 1, 9, 3, 7, 11, 1, 1, 47, 5, 31, 15, 1, 1, 11, 193, 3, 103, 3, 1, 8807, 1, 3383, 3, 21, 3, 8005, 1, 1, 13, 17, 3, 2047

Offset: 1

Author

Wolfdieter Lang, Oct 29 2015

Keywords

• nonn

Comments

The 2-happy numbers D(n)*E(n) are given in A007970(n) (called rhombic numbers in the Conway paper). D(n) = A191856(n), E(n) = A191857(n). Here the corresponding smallest positive numbers satisfying E(n)*U(n)^2 - D(n)*T(n)^2 = +2, n >= 1, with odd U(n) and T(n) are given as T0(n) = a(n) and U0(n) = A263009(n).

In the W. Lang link the first U0(n) and T0(n) numbers are given in the Table for d(n) = A007970(n), n >= 1.

In the Zumkeller link "Initial Happy Factorization Data" given in A191860 the a(n) = T0(n) numbers appear for the t = 2 rows in column v.

Examples

			n = 6: 2-happy number A007970(6) = 19 = 1*19 = A191856(6)*A191857(6). 19*A263009(6)^2 - 1*a(6)^2 = 19*3^2 - 1*13^2 = +2. This is the smallest positive solution for the given 2-happy couple (A191856(n), A191857(n)).
		

Links

• J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1.
• Wolfdieter Lang, Proof of a Theorem Related to the Happy Number Factorization.

Crossrefs

Cf. A007970, A191856, A191857, A191860, A263009, A262026, A262027, A262028.

Formula

A191857(n)*A263009(n)^2 - A191856(n)*a(n)^2 = +2, and a(n) with A263009(n) is the smallest positive solution for the given 2-happy couple (A191856(n), A191857(n)).